Classification of Euclidean signature metrics admitting a rank 2 closed conformal Killing-Yano tensor

In this thesis, we start with a review of topics such as Lie derivatives, Killing vectors, and Killing tensors, as well as the generalizations of these objects to their conformal variants.We then introduce the form of the Kerr-NUT-(A)dS metric for an arbitrary number of dimensions, followed by a review and proof of a result by Houri, Oota, and Yasui.The main takeaway from this result is that if a metric of Euclidean signature, satisfying the Einstein equations, admits a rank 2 closed conformal Killing-Yano tensor, then it must be a member of the Kerr-NUT-(A)dS family of metrics.At the end, more recent work done by Frolov, Krtous, and Kubiznak is briefly mentioned, where a new family of metrics (not of Euclidean signature) is found to admit a principal tensor; a non-degenerate rank 2 closed conformal Killing-Yano tensor.This leads to the belief that the classification of metrics which admit these types of Killing-Yano tensors is far from complete.i this, it was found that the geodesic equation of Kerr was indeed integrable, although the exact reason was not completely discerned until later.Killing vectors and Killing tensors can be further generalized into objects known as conformal Killing vectors and conformal Killing tensors.Beyond that, we can generalize tensors even further into objects known as Killing-Yano tensors, initially described by Yano in 1952 [5].These Killing-Yano tensors turned out to be the key for furthering our understanding of integrable spacetimes (by integrable spacetimes here, we mean spacetimes in which the fundamental wave equations, i.e., the Klein-Gordon, Dirac, and Hamilton-Jacobi equations, can be solved by separation of variables).It turns out that the existence of a special Killing-Yano tensor, referred to as the principal conformal Killing-Yano tensor, is helpful in defining this large class of integrable spacetimes.In this thesis, we will discuss the most general (Euclidean signature) metric that solves the Einstein equations which admits a rank 2 closed conformal Killing-Yano tensor: the D-dimensional Kerr-NUT-(A)dS metric.We will start by giving a review of the objects mentioned above: Killing vectors, Killing tensors, Killing-Yano tensors, etc., in chapter 2. In chapter 3 we will introduce the Kerr-NUT-AdS family of metrics in their most general forms, as described in [10].In chapter 4, we will go over the proof of the theorem originally proven by Houri, Oota, and Yasui [16] which (loosely) states that if we have a D-dimensional spacetime (M, g) with rank 2, closed conformal Killing-Yano tensor h with some assumption on the structure of its eigenvalues, then we can deduce a form for the metric g and the Killing-Yano tensor h.The caveat of this result is that, when imposing the Einstein equations on the resulting metric g, it turns out that the only metric which satisfies both conditions simultaneously is the Kerr-NUT-(A)dS metric.The proof of the theorem in [16] is broken down into several lemmas which we will first state and then provide proofs.The proof assumes that the metrics in question are of Euclidean signature.However, in recent years, Frolov, Krtous, and Kubiznak [17] have been able to uncover metrics of different signature (i.e., Lorentzian) which admit a principal tensor, the non-degenerate analogue of the rank 2 closed conformal Killing-Yano tensor in question.This leads use to believe that the problem of classifying these metrics which admit these tensors is far from complete, and what we know as of now is only a small step towards a complete classification.In chapter 5, we will remark a bit more on this recent work, and end with a summary of information presented and some closing remarks.

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